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The paper addresses an open problem raised in [Bartsch, Molle, Rizzi,Verzini: Normalized solutions of mass supercritical Schr\"odinger equationswith potential, Comm. Part. Diff. Equ. 46 (2021), 1729-1756] on the existenceof normalized solutions to Schr\"odinger equations with potentials andinhomogeneous nonlinearities. We consider the problem \[  -\Delta u+V(x)u+\lambda u = |u|^{q-2}u+\beta |u|^{p-2}u, \quad\|u\|^2_2=\int|u|^2dx = \alpha, \] both on $\mathbb{R}^N$ as well as on domains$r\Omega$ where $\Omega\subset\mathbb{R}^N$ is an open bounded convex domainand $r>0$ is large. The exponents satisfy $2<p<2+\frac4N<q<2^*=\frac{2N}{N-2}$,so that the nonlinearity is a combination of a mass subcritical and a masssupercritical term. Due to the presence of the potential a by now standardapproach based on the Pohozaev identity cannot be used. We develop a robustmethod to study the existence of normalized solutions of nonlinearSchr\"odinger equations with potential and find conditions on $V$ so thatnormalized solutions exist. Our results are new even in the case $\beta=0$.

Normalized solutions to Sch\"odinger equations with potential and inhomogeneous nonlinearities on large convex domains